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Answer :
To evaluate the logarithmic expression [tex]\(\log_5 \frac{1}{25}\)[/tex], we want to determine the exponent to which 5 must be raised to get [tex]\(\frac{1}{25}\)[/tex].
1. Understand the expression: We are looking for the value of [tex]\(x\)[/tex] in the equation:
[tex]\[
5^x = \frac{1}{25}
\][/tex]
2. Express [tex]\(\frac{1}{25}\)[/tex] as a power of 5:
Since [tex]\(25 = 5^2\)[/tex], we can express [tex]\(\frac{1}{25}\)[/tex] as the reciprocal of a power of 5:
[tex]\[
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}
\][/tex]
3. Relate this back to the logarithmic equation:
Now, from the original equation [tex]\(5^x = \frac{1}{25}\)[/tex], we see that:
[tex]\[
5^x = 5^{-2}
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Since the bases are the same (both are 5), we can equate the exponents:
[tex]\[
x = -2
\][/tex]
Therefore, the value of [tex]\(\log_5 \frac{1}{25}\)[/tex] is [tex]\(-2\)[/tex].
So, the correct answer is:
A. [tex]\(\log_5 \frac{1}{25} = -2\)[/tex]
1. Understand the expression: We are looking for the value of [tex]\(x\)[/tex] in the equation:
[tex]\[
5^x = \frac{1}{25}
\][/tex]
2. Express [tex]\(\frac{1}{25}\)[/tex] as a power of 5:
Since [tex]\(25 = 5^2\)[/tex], we can express [tex]\(\frac{1}{25}\)[/tex] as the reciprocal of a power of 5:
[tex]\[
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}
\][/tex]
3. Relate this back to the logarithmic equation:
Now, from the original equation [tex]\(5^x = \frac{1}{25}\)[/tex], we see that:
[tex]\[
5^x = 5^{-2}
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Since the bases are the same (both are 5), we can equate the exponents:
[tex]\[
x = -2
\][/tex]
Therefore, the value of [tex]\(\log_5 \frac{1}{25}\)[/tex] is [tex]\(-2\)[/tex].
So, the correct answer is:
A. [tex]\(\log_5 \frac{1}{25} = -2\)[/tex]
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